# What is Fourier transform: Definition + 1000 Threads

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.
The Fourier transform of a function of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation, as proven by the Fourier inversion theorem.

Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint. The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or Rn (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.

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1. ### I Fourier transform of the density fluctuation

There is a Fourier transform that I don't really understand in my textbook. I have the following equation: ##\ddot{\delta} + 2H\dot{\delta} -\frac{3}{2} \Omega_m H^2 \delta = 0## Then using the Fourier transform: ##\delta_{\vec{k}} = \frac{1}{V} \int \delta(\vec{r}) e^{i \vec{k} \cdot \vec{r}}...
2. ### I New frequency generation in AM signal

Suppose I have a pure sine wave. Upon Fourier transforming (FT) the time signal, I obtain a delta function in the frequency domain. If I subsequently modulate this sine wave with another function, for instance, a Gaussian, the delta function in the frequency domain will broaden. I'm curious...
3. ### Fourier Transform vs Short Time Fourier Transform...

Hello, I understand how the FT and the STFT work. The STFT provides time-frequency localization, i.e. it can tell us when the spectral components are acting in the time-domain signal...The STFT is also useful for non-stationary signals which are signals whose statistical characteristics are...
4. ### Showing that a certain summation is equal to a Dirac delta?

I'm studying Quantum Field Theory for the Gifted Amateur and feel like my math background for it is a bit shaky. This was my attempt at a derivation of the above. I know it's not rigorous, but is it at least conceptually right? I'll only show it for bosons since it's pretty much the same for...
5. ### Model CO2 diffusing across the wall of a cylindrical alveolar blood vessel

TL;DR Summary: Solve heat equation in a disc using fourier transforms Carbon dioxide dissolves in the blood plasma but is not absorbed by red blood cells. As the blood returns to an alveolus, assume that it is well-mixed so that the concentration of dissolved CO2 is uniform across a...

11. ### Fourier transform of ##e^{-a |t|}\cos{(bt)}##

First, ##\tilde{f}(\omega)=\int_{-\infty}^{\infty}e^{a|t|}\cos(bt)e^{-i\omega t} \mathrm{d}t## We can get rid of the absolute value by splitting the integral up ##\int_{-\infty}^{0}e^{at}\cos(bt)e^{-i\omega t} \mathrm{d}t+ \int_{0}^{\infty}e^{-at}\cos(bt)e^{-i\omega t} \mathrm{d}t## Using...
12. ### Fourier transform of wave packet

I am unsure if ##h(x,t)## really is a wave packet, but it looks like one, hence the title. Anyway, so I'd like to determine ##\hat{h}(k,t=0)##. My attempt so far is recognizing that, without the real part in the integral, i.e. ##g(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} a(k)e^{i(kx-\omega...
13. ### A Polar Fourier transform of derivatives

The 2D Fourier transform is given by: \hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy In terms of polar co-ordinates: \hat{f}(\rho,\phi)=\int_{0}^{\infty}\int_{-\pi}^{\pi}rf(r,\theta)e^{-ir\rho\cos(\theta-\phi)}drd\theta For Fourier transforms in cartesian co-ordinates, relating the...
14. ### "Simple" Fourier transform problem

I am unsure about what is being asked for in the question. At first I thought the question asks one to calculate the inverse Fourier transform and then to analyze its depends on ##t##, however, the "estimate" makes me think otherwise.
15. ### Simple Fourier transformation calculation

So, ##\hat{p}(\omega)=\int_{-\infty}^{\infty} p(t)e^{-i\omega t}\mathrm{d}t=A\int_{0}^{\infty}e^{-t(\gamma+i(\omega+\omega_0))}=A\left[-\frac{e^{-t(\gamma+i(\omega+\omega_0))}}{\gamma+i(\omega+\omega_0)}\right]_0^\infty,## provided ##\gamma+ i(\omega+\omega_0)\neq 0## for the last equality. Now...

36. ### I Does Each Component of a Vector Have an Independent Fourier Transform?

Given ##f(\vec{x})##, where the Fourier transform ##\mathcal{F}(f(\vec{x}))= \hat{f}(\vec{k})##. Given ##\vec{x}=[x_1,x_2,x_3]## and ##\vec{k}=[k_1,k_2,k_3]##, is the following true? \begin{split} \mathcal{F}(f(x_1))&= \hat{f}(k_1) \\ \mathcal{F}(f(x_2))&= \hat{f}(k_2) \\...
37. ### I Get the time axis right in an inverse Fast Fourier Transform

Hi I would like to transform the S-parameter responce, collected from a Vector Network Analyzer (VNA), in time domain by using the Inverse Fast Fourier Transform (IFFT) . I use MATLAB IFFT function to do this and the response looks correct, the problem is that I do not manage to the time scaling...
38. ### Fourier transform of electric susceptibility example

I have not studied the Fourier transform (FT) in great detail, but came across a problem in electrodynamics in which I assume it is needed. The problem goes as follows: Evaluate ##\chi (t)## for the model function...
39. ### The Fast Fourier Transform is described in the Quantum Domain

In August, "Quantum Information Processing" published an article describing a full FFT in the quantum domain - a so-called QFFT, not to be confused with the simpler QFT. According to the publication:
40. ### Mathematica Fourier Transform Help with Mathematica

I am attempting to be able to do this problem with the help of Mathematica and Fourier transform. My professor gave us instructions for Fourier Transformation and Inverse Fourier, but I don't believe that my output in Mathematica is correct.
41. ### I Why should a Fourier transform not be a change of basis?

I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts. It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...
42. ### Engineering Fourier Transform: best window to represent function

Hi, I was hoping to gain more insight into these window questions when looking at frequency spectra questions. I don't really know what makes windows better than one another. My attempt: In the question, we have f(t) = cos(\omega_0 t) and therefore its F.T is F(\omega ) = \pi \left(...
43. ### Fourier transform of Maxwell's equations

Hello, I am unfamiliar with Maxwell's equations' Fourier transform. Are there any materials talking about it?

45. ### I Linearity of power spectral density calculations

I have a question related to linearity of power spectral density calculation. Suppose I have a time series, divided into some epochs. If I compute PSD by Welch's method with a time window equal to the length of an epoch and without any overlap, I obtain this result: If I calculate the...
46. ### What is the Exponential Fourier Transform of an Even Function?

From the sketch, I know that this function is an even function. So, I simplify the Fourier transform in the limit of the integration (but still in exponential form). Then, I try to find the exponential FOurier transform. Here what I get: g(a)=\frac{2}{2\pi} \int_{0}^{\infty} e^{-x} e^{-iax}...
47. ### I Equivalence of two different definitions of quasicrystals

https://en.wikipedia.org/wiki/Riemann_hypothesis#Quasicrystals a quasicrystal as "a distribution with discrete support whose Fourier transform also has discrete support." https://en.wikipedia.org/wiki/Quasicrystal#Mathematicsdefines a quasicrystal as "a structure that is ordered but not...
48. ### Fourier series and the shifting property of Fourier transform

Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform. So here's my attempt to this problem so far...
49. ### Finding Fourier Transforms of Non-Rectangular Pulses

Hi, In class I have learned how to find the Fourier transform of rectangular pulses. However, how do I solve a problem when I should sketch the Fourier transform of a pulse that isn't exactly rectangular. For instance "Sketch the Fourier transform of the following 2 pulses" Thanks in advance...
50. ### I The domain of the Fourier transform

Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex? For example, given \begin{split} \hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...